We study the roots of the generalised Hermite polynomials H _m,n when both m and n are large. We prove that the roots, when appropriately rescaled, densely fill a bounded quadrilateral region, called the elliptic region, and organise themselves on a deformed rectangular grid, as was numerically observed by Clarkson. We describe the elliptic region and the deformed grid in terms of elliptic integrals and their degenerations.

中文翻译：

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当两个参数都很大时，广义 Hermite 多项式的根

当m和n都很大时_，我们研究了广义 Hermite 多项式H _{m , n}的根。我们证明，当适当地重新缩放时，根会密集地填充一个称为椭圆区域的有界四边形区域，并将它们自己组织在变形的矩形网格上，正如克拉克森在数值上观察到的那样。我们根据椭圆积分及其退化来描述椭圆区域和变形网格。